Solve simple rational and radical equations and identify extraneous solutions.
Solve a rational equation by clearing denominators and checking restrictions
Problem
Solve \(1/x~+~1/(x~+~1)~=~1/2\). State the solution set and all original denominator restrictions.
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What this problem is really about
Begin by listing every value that makes an original denominator zero. Build one least common denominator containing every needed factor, and multiply every term on both sides by it; missing even one term changes the equation. Solve the polynomial that remains, keep exact forms when possible, and compare every candidate with the original restrictions before accepting it.
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